Average Error: 60.0 → 0.0
Time: 26.5s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{1}{9} + \left(x \cdot \left(\frac{1}{45} \cdot x\right) - \frac{1}{3}\right) \cdot \left(x \cdot \left(\frac{1}{45} \cdot x\right)\right)}{\left(x \cdot \left(\frac{1}{45} \cdot x\right)\right) \cdot \left(\left(x \cdot \left(\frac{1}{45} \cdot x\right)\right) \cdot \left(x \cdot \left(\frac{1}{45} \cdot x\right)\right)\right) + \frac{1}{27}}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{1}{9} + \left(x \cdot \left(\frac{1}{45} \cdot x\right) - \frac{1}{3}\right) \cdot \left(x \cdot \left(\frac{1}{45} \cdot x\right)\right)}{\left(x \cdot \left(\frac{1}{45} \cdot x\right)\right) \cdot \left(\left(x \cdot \left(\frac{1}{45} \cdot x\right)\right) \cdot \left(x \cdot \left(\frac{1}{45} \cdot x\right)\right)\right) + \frac{1}{27}}}
double f(double x) {
        double r1902339 = 1.0;
        double r1902340 = x;
        double r1902341 = r1902339 / r1902340;
        double r1902342 = tan(r1902340);
        double r1902343 = r1902339 / r1902342;
        double r1902344 = r1902341 - r1902343;
        return r1902344;
}
↓
double f(double x) {
        double r1902345 = x;
        double r1902346 = 5.0;
        double r1902347 = pow(r1902345, r1902346);
        double r1902348 = 0.0021164021164021165;
        double r1902349 = r1902347 * r1902348;
        double r1902350 = 0.1111111111111111;
        double r1902351 = 0.022222222222222223;
        double r1902352 = r1902351 * r1902345;
        double r1902353 = r1902345 * r1902352;
        double r1902354 = 0.3333333333333333;
        double r1902355 = r1902353 - r1902354;
        double r1902356 = r1902355 * r1902353;
        double r1902357 = r1902350 + r1902356;
        double r1902358 = r1902353 * r1902353;
        double r1902359 = r1902353 * r1902358;
        double r1902360 = 0.037037037037037035;
        double r1902361 = r1902359 + r1902360;
        double r1902362 = r1902357 / r1902361;
        double r1902363 = r1902345 / r1902362;
        double r1902364 = r1902349 + r1902363;
        return r1902364;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.0
Target0.1
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 60.0

    \[\frac{1}{x} - \frac{1}{\tan x}\]

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\frac{1}{3} \cdot x + \left(\frac{1}{45} \cdot {x}^{3} + \frac{2}{945} \cdot {x}^{5}\right)}\]

  3. Simplified0.3

    \[\leadsto \color{blue}{{x}^{5} \cdot \frac{2}{945} + x \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \frac{1}{45}\right)\right)}\]
  4. Using strategy rm

  5. Applied flip3-+1.2

    \[\leadsto {x}^{5} \cdot \frac{2}{945} + x \cdot \color{blue}{\frac{{\frac{1}{3}}^{3} + {\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right)}^{3}}{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right)\right)}}\]

  6. Applied associate-*r/1.1

    \[\leadsto {x}^{5} \cdot \frac{2}{945} + \color{blue}{\frac{x \cdot \left({\frac{1}{3}}^{3} + {\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right)}^{3}\right)}{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right)\right)}}\]

  7. Simplified0.3

    \[\leadsto {x}^{5} \cdot \frac{2}{945} + \frac{\color{blue}{x \cdot \left(\frac{1}{27} + \left(\left(x \cdot \frac{1}{45}\right) \cdot x\right) \cdot \left(\left(\left(x \cdot \frac{1}{45}\right) \cdot x\right) \cdot \left(\left(x \cdot \frac{1}{45}\right) \cdot x\right)\right)\right)}}{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right)\right)}\]
  8. Using strategy rm

  9. Applied associate-/l*0.0

    \[\leadsto {x}^{5} \cdot \frac{2}{945} + \color{blue}{\frac{x}{\frac{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right)\right)}{\frac{1}{27} + \left(\left(x \cdot \frac{1}{45}\right) \cdot x\right) \cdot \left(\left(\left(x \cdot \frac{1}{45}\right) \cdot x\right) \cdot \left(\left(x \cdot \frac{1}{45}\right) \cdot x\right)\right)}}}\]

  10. Simplified0.0

    \[\leadsto {x}^{5} \cdot \frac{2}{945} + \frac{x}{\color{blue}{\frac{\left(\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) + \frac{1}{9}}{\left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \left(\left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right)\right) + \frac{1}{27}}}}\]

  11. Final simplification0.0

    \[\leadsto {x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{1}{9} + \left(x \cdot \left(\frac{1}{45} \cdot x\right) - \frac{1}{3}\right) \cdot \left(x \cdot \left(\frac{1}{45} \cdot x\right)\right)}{\left(x \cdot \left(\frac{1}{45} \cdot x\right)\right) \cdot \left(\left(x \cdot \left(\frac{1}{45} \cdot x\right)\right) \cdot \left(x \cdot \left(\frac{1}{45} \cdot x\right)\right)\right) + \frac{1}{27}}}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "invcot (example 3.9)"
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

  (- (/ 1 x) (/ 1 (tan x))))